Fractional Differentiation Through Bohr Spectra: Regularity, Inversion, and Memory Effects

This paper develops a Bohr‐spectral framework for fractional differentiation of Banach‐valued almost periodic functions. A closed fractional differentiation operator is defined on a maximal spectral domain by the multiplier ( i λ ) ^α acting on Bohr–Fourier coefficients. Weighted Wiener almost periodic classes are then used to obtain explicit fractional regularity estimates and to identify conditions for composition and inversion, with particular attention to the low‐frequency difficulty arising in Wiener‐class inversion when nonzero frequencies accumulate at the origin. The translation‐invariant spectral realization is subsequently compared with the finite‐terminal Caputo derivative for 0 < α < 1. Under the sufficient regularity assumption , an exact decomposition is established into a persistent almost periodic component and a terminal‐memory transient. The latter is shown to decay at the algebraic rate O (( t − a ) ^{− α} ), yielding a unique asymptotically almost periodic decomposition and recovery of the persistent nonzero Bohr spectrum. A single‐frequency analysis further proves that finite‐terminal Caputo differentiation does not preserve Bohr almost periodicity in general. Quasiperiodic and low‐frequency examples illustrate the spectral action, the decay of terminal memory, and the limitation of Wiener‐class fractional inversion when nonzero frequencies accumulate near the origin.